I have always loved the discrete and hated the continuous. Perhaps it was the trauma of having to go through the usual curriculum of “rigorous ” , Cauchy-Weierstrass-style, real calculus, where one has all those tedious, pedantic and utterly boring, ɛ − δ proofs. The meager (obvious) conclusions hardly justify the huge mental efforts! Complex Analysis was a different story. Even though officially “continuous”, it has the feel of discrete math, and one can “cheat ” and consider power series as formal power series, and I really loved it. Georgy P. Egorychev: A Bridge-Builder between the Discrete and the Continuous Eight years after I finished my doctorate, I came across Egorychev’s fascinating modern classic[E], about using the methods of complex analysis to evaluate (discrete) combinatorial sums. That was a pioneering ecumenical work, that influenced me greatly. Its content, of course, but especially its spirit and philosophy. The Discrete vs. The Continuous: A Two-Way Street Egorychev went from the discrete to the continuous. But the bridge that he helped build can be transversed both ways. With the advent of so-called Wilf-Zeilberger (WZ) theory[WZ] one can

Aristotelian, or non-Platonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity,

We introduce and discuss a concept of approximation of a topological algebraic system A by finite algebraic systems from a given class K. If A is discrete, this concept agrees with the familiar notion of a local embedding of A in a class K of algebraic systems. One characterization of this concept states that A is locally embedded in K iff it is a subsystem of an ultraproduct of systems from K. In this paper we obtain a similar characterization of approximability of a locally compact system A by systems from K using the language of nonstandard analysis. In the signature of A we introduce positive bounded formulas and their approximations; these are similar to those introduced by Henson [14] for Banach space structures (see also [15, 16]). We prove that a positive bounded formula ϕ holds in A if and only if all precise enough approximations of ϕ hold in all precise enough approximations of A. We also prove that a locally compact field cannot be approximated arbitrarily closely by finite (associative) rings (even if the rings are allowed to be non-commutative). Finite approximations of the field R can be considered as possible computer systems for real arithmetic. Thus, our results show that there do not exist arbitrarily accurate computer arithmetics for the reals that are associative rings. 1